You can have third parties under FPTP. The UK, Canada, India to name a few have them.

So does the US, they even occasional win elections, even to Congress (historically, there is at least one case where one displaced a major national party in the state-level duopoly), that doesn't mean FPTP doesn't tend toward duopoly. (FPTP plus a strong Presidential system tends more strongly toward duopoly; FPTP plus a strong Presidential system plus an indirectly-elected President where Presidential electors are mostly elected by multimember FPTP tends even more strongly to duopoly.)

Isn't it more of a property of single-member districts rather than voting system?

No, multimember FPTP districts would be the same but worse (that's pretty much what most states use for Presidential electors, though), whereas you could construct a single-member district system with an electoral system producing proportionality and supporting multiple parties (though it would probably be ugly to administer) [0] which would not have the same problem (though it might have all kinds of other problems). It's a absolutely an FPTP problem, though most reasonable solutions also involve multimember districts.

[0] for an example of the top of my head: vote as in single-member FPTP, determine the proportion of first place votes globally by party to determine how many seats each party must win, and then globally order candidates from each party entitled to at least one seat in descending order of their vote share in their district. Starting with the party with the candidate with lowest maximum vote share, assign that candidate as the winner from their district, removing all other candidates from that district and dropping the party from further consideration if it has been assigned the number of seats it is entitled to; proceed until all seats are assigned.

> determine the proportion of first place votes globally

This sounds to me like artificially creating an election-wide multi-member district. I don't think it's standard to allow votes outside of a particular voting district to affect the results in that voting district.

I'm not claiming that making districts multi-member is a solution (as you mention, multi-member, winner-takes-all districts are also bad), just that it is a necessary condition to arrive at results that are closer to representing the choice of the voters.

Edit: apologies if this feels like moving the goal-post. I need to think a bit more about what could happen for example in single-member voting district with ranked-choice voting system - I think it would not lead to a 2-party system, so you're right here and changing the voting system may be sufficient.

> This sounds to me like artificially creating an election-wide multi-member district

I can see that.

> I'm not claiming that making districts multi-member is a solution (as you mention, multi-member, winner-takes-all districts are also bad), just that it is a necessary condition to arrive at results that are closer to representing the choice of the voters.

Ranked choice methods with single-member districts can slightly mitigate some of the problems, but yes, I’d agree that realistic solutions that do more than nip around the edges require multimember districts.